Based on empirical evidence of fast mean-reverting spikes, electricity spot prices are often modeled $X+Z^\beta$ as the sum of a continuous It\^o semimartingale $X$ and a mean-reverting compound Poisson process $Z_t^\beta = \int_0^t\int_{\R} xe^{-\beta(t-s)}\underline{p}(ds,dt)$ where $\underline{p}(ds,dt)$ is Poisson random measure with intensity $\lambda ds\otimes dt$. In a first part, we investigate the estimation of $(\lambda,\beta)$ from discrete observations and establish asymptotic efficiency in various asymptotic settings. In a second part, we discuss the use of our inference results for correcting the value of forward contracts on electricity markets in presence of spikes. We implement our method on real data in the French, German and Australian market over 2015 and 2016 and show in particular the effect of spike modelling on the valuation of certain strip options. In particular, we show that some out-of-the-money options have a significant value if we incorporate spikes in our modelling, while having a value close to $0$ otherwise.
We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. We derive two models based on the structure of the Reflection Brownian Copula which present two states of correlation ; one is directly based on the reflection of the Brownian motion and the other is a local correlation model. These models can be used for risk management and option pricing in commodity energy markets. (2010): 60J25, 60J60, 60J65, 60J70, 60H10, 62H99.
Mathematics Subject Classification
We consider a 2-dimensional marked Hawkes process with increasing baseline intensity in order to model prices on electricity intraday markets. This model allows to represent different empirical facts such as increasing market activity, random jump sizes but above all microstructure noise through the signature plot. This last feature is of particular importance for practitioners and has not yet been modeled on those particular markets. We provide analytic formulas for first and second moments and for the signature plot, extending the classic results of Bacry et al. [2] in the context of Hawkes processes with random jump sizes and time dependent baseline intensity. The tractable model we propose is estimated on German data and seems to fit the data well. We also provide a result about the convergence of the price process to a Brownian motion with increasing volatility at macroscopic scales, highlighting the Samuelson effect.
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